Constant Rate Codes with Fully Addressable Transversal T: Good Codes, Sparse Checks

Abstract:Recent work by Wills constructed codes with constant rate and transversal $T$. Extending this line of work, we construct asymptotically good CSS codes with \textbf{fully addressable} transversal $T$, permitting selective control over which logical qubits receive a $T$ gate. We also give Pauli-LDPC codes with transversal $T$ and growing distance at vanishing rate; and a protected-subsystem construction with fully addressable transversal $T$, constant protected rate, distance $\Omega(\sqrt n)$, and sparse (non-Pauli) checks. More generally, we obtain asymptotically good binary codes with full transversal addressability for every fixed finite-order one-qubit gate, retaining CSS realizations for dyadic phase gates such as $\sqrt T$ and using generally nonadditive codes for rotations such as $R_z(\pi/7)$. The remaining challenge is to construct asymptotically good Pauli-LDPC codes with fully addressable transversal non-Clifford one-qubit gates.

arXiv:2609.38996 [quant-ph]

(or arXiv:2609.38996v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.38996 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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